(4x%2B3)%3D112.jpeg "(2x-7)(4x+3)=112")
Solving the Equation: (2x - 7)(4x + 3) = 112
This article will guide you through the steps of solving the equation (2x - 7)(4x + 3) = 112.
Expanding the Equation
Firstly, we need to expand the left side of the equation by using the distributive property (or FOIL method):
(2x - 7)(4x + 3) = 112
- 2x * 4x = 8x²
- 2x * 3 = 6x
- -7 * 4x = -28x
- -7 * 3 = -21
Combining these terms, we get:
8x² - 22x - 21 = 112
Rearranging the Equation
To solve for 'x', we need to set the equation to zero:
8x² - 22x - 21 - 112 = 0
8x² - 22x - 133 = 0
Solving the Quadratic Equation
Now we have a quadratic equation in the form of ax² + bx + c = 0. There are several methods to solve this:
- Factoring: In this case, factoring might be difficult.
- Quadratic Formula: This is the most reliable method for any quadratic equation.
Quadratic Formula:
x = (-b ± √(b² - 4ac)) / 2a
Where:
- a = 8
- b = -22
- c = -133
Substituting these values into the formula:
x = (22 ± √((-22)² - 4 * 8 * -133)) / (2 * 8)
x = (22 ± √(484 + 4256)) / 16
x = (22 ± √(4740)) / 16
x = (22 ± 2√1185) / 16
Therefore, the solutions for the equation (2x - 7)(4x + 3) = 112 are:
- x = (22 + 2√1185) / 16
- x = (22 - 2√1185) / 16
These solutions can be further simplified or approximated to decimal values.